#7021824

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996607
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#59129	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2325 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,19,17,22,15,21)(16,24,25,28,26,20,23,27);(5,6)(7,11)(8,10)(9,12)(14,19)(15,18)(16,23)(21,22)(25,26)(27,28);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15,17,18)(14,19,22,21)(16,25,26,23)(20,28,24,27);(7,12)(9,11)(13,16,15,23,17,26,18,25)(14,20,21,28,22,24,19,27);(6,10)(9,11)(13,15,17,18)(14,21,22,19)(16,25,26,23)(20,27,24,28)

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6929020);